Papers

In this paper, we propose Gestalt, a new paradigm of large multimodal model built around multimodal interplay. Despite rapid advances, large multimodal models are reaching a bottleneck: existing approaches focus primarily on accommodating additional modalities while overlooking the distinct characteristics of each modality and the relations among them. Motivated by the multistage property of human multisensory perception, we propose a multimodal interplay pyramid that organizes multimodal modeling as a progression from modality-specific processing, through cross-modal alignment, to deeper multimodal integration. Guided by this pyramid, Gestalt adopts a unified discrete diffusion framework and an interplay-partitioned architecture, with learnable interplay tokens mediating cross-modal exchange and integration. The pyramid also structures its data organization and training strategy. Strong performance across image generation, multimodal understanding, and text-only evaluation shows that Gestalt significantly improves cross-modal integration while preserving modality-specific information, effectively harnessing the strengths of diffusion-based multimodal models and offering a promising path toward unified multimodal intelligence. Project page: https://GeWu-Lab.github.io/Gestalt.

A common approach to world-model simulation for vision-language-action (VLA) systems is to predict future RGB observations and then re-encode them into policy inputs, introducing an indirect interface between simulation and downstream policy execution. We instead investigate whether world dynamics can be modeled in a compact, policy-oriented state derived from VLM visual tokens. A key challenge is that raw VLM visual tokens are high-dimensional, making efficient and accurate autoregressive dynamics modeling challenging. To address this, we introduce Token-World, an action-conditioned world model that compresses VLM features into a compact token state, learns future dynamics in this reduced space, and maps predicted states back to the original policy-facing representation for downstream use. Across manipulation benchmarks, Token-World improves open-loop feature fidelity and policy-action consistency over recent world-model simulators, with slower degradation over long rollout horizons. In closed-loop evaluation, its simulated policy performance correlates more strongly with reference policy performance than Ctrl-World ($r=0.794$ vs.\ $0.583$), while requiring lower simulation latency. Ablations further show that compact-representation design and dimensionality substantially affect future-state prediction. Code will be available at https://chuyaofu.github.io/Token-World/.

Multi-reward reinforcement learning trains large language models to satisfy multiple behavioral objectives simultaneously. Reward-wise normalization, as used in GDPO, preserves reward-specific relative information within rollout groups, but different objectives can still exhibit uneven learning progress. We study this behavior through advantage energy, the sum of a reward's squared advantages over a batch. Under idealized GDPO normalization, we show that this energy is proportional to active-group density: the fraction of rollout groups in which the reward provides nonzero relative advantages. This reveals a residual batch-level signal imbalance and provides a basis for calibrating reward contributions. Based on this relation, we propose Density-Aware Reward Aggregation (DARA). We derive an inverse-square-root density correction that gives greater weight to signals from less frequently active rewards. DARA computes its weights from each rollout batch, adapting to changes in reward activity throughout training without modifying the underlying policy optimization objective. Experiments on tool calling and mathematical reasoning show that DARA learns the targeted behaviors faster than GDPO, reaching high format compliance in up to 26% fewer training steps on tool calling and near-saturated length compliance in up to 65% fewer steps on mathematical reasoning, while remaining competitive in final performance. Our code is available at https://github.com/zhaihaotian/DARA.

Query-aware keyframe selection enables multimodal large language models (MLLMs) to process long videos using only a small set of question-relevant frames. Existing score-based methods, however, typically search within a fixed, uniformly sampled candidate pool, preventing evidence outside this pool from ever being selected. Given a limited relevance-scoring budget, the key challenge is to allocate evaluations adaptively to promising frames while continuing to explore underrepresented temporal regions. We introduce FORTE, a training-free framework that addresses this challenge through two stages: adaptive relevance scoring and global keyframe optimization. Starting from sparse, uniformly distributed observations, our efficient Gaussian-process relevance predictor estimates relevance for unscored frames, exploiting temporal locality and the approximately banded kernel structure to reduce the core computation from cubic to linear time in the number of frames for fixed bandwidth. The scoring stage then selects which frames to score next by balancing predicted relevance with temporal coverage, prioritizing promising regions while also exploring less-represented parts of the video. The optimization stage selects the final keyframes by maximizing an objective that jointly captures measured relevance and temporal coverage. We derive an exact algorithm that leverages the logarithmic coverage structure to identify the optimal subset of the scored candidate pool in time linear in the pool size, for a fixed final-frame budget. Experiments on four long-video question-answering benchmarks show that FORTE achieves the highest observed mean accuracy among the compared selectors under every tested scoring budget. Further evaluations demonstrate its consistent effectiveness across different relevance scorers and downstream MLLMs.

Motivated by John Baez's question about seven exceptional coincidences among binomial coefficients, we construct seven exterior-power identities for representations of $A_5$. Their common symmetry comes from simplex edges and icosahedral vertices, axes, and faces, together with characters of stabilizers. We enumerate all ordered pairs of representation isomorphism classes in the prescribed dimensions. Apart from the pair of trivial actions, the permutation pair at $1540$ and the complex monomial pair at $7140$ are unique. At $3003$, Klein's twelve-point divisor gives a noncanonical intertwiner factored through polynomial division, differentiation, and residues.

Large reasoning models (LRMs) have achieved substantial improvements in solving complex mathematical problems, but often produce lengthy, repetitive, or erroneous reasoning trajectories. In this work, we introduce a new interpretability framework, SLIDER, to evaluate the quality of the reasoning process. SLIDER leverages an emerging body of work from information theory called Partial Information Decomposition to disentangle the information about the final answer between two consecutive reasoning steps into non-negative components: unique information (in preceding steps or current step), redundant information, and synergistic information. Building on this decomposition, we propose the *Step-wise Repetitive Reasoning Index (Step-RRI)*, a theoretically grounded measure that assesses whether the answer-relevant information in the current step $S_i$ is predominantly redundant with the past steps $S_{<i}$, relative to its unique and synergistic contributions. To evaluate the effectiveness of Step-RRI in detecting repetitiveness, we apply SLIDER to the redundancy class of the PRMBench dataset where Step-RRI improves step-level redundancy identification accuracy by over $10$ points compared to embedding-similarity and information-gain baselines. Next, we define *Trajectory-RRI*, an aggregate measure of repetitiveness for an individual reasoning trajectory. To demonstrate its practical relevance, we show that average Trajectory-RRI strongly correlates with actual reasoning length across QwQ-32B, DeepSeek-R1-Distill-Qwen-32B, and GPT-4.1, motivating its use as a signal for improving reasoning efficiency. Finally, we introduce *Trajectory-RRI-guided data selection for fine-tuning*, demonstrating that selecting training data based on Trajectory-RRI can improve a fine-tuned model's reasoning efficiency while largely preserving its task performance.

The OrcaQuest CMOS sensor by Hamamatsu Photonics has been found to exhibit a non-linear gain curve at low exposure times. This effect is characterized through the various exposure tests and data reduction pipelines detailed in this report, and a more accurate gain characterization curve is provided. Such data can be used to provide more consistent measures of pixel gain, linearity, and high framerate images.

Full detector simulation and reconstruction of collider events are projected to become major bottlenecks at the High-Luminosity Large Hadron Collider, motivating the development of fast, ML-based surrogates. At the same time, LLMs have driven fast progress in generative discrete modeling: autoregressive transformers trained on tokenized data now represent the state of the art across a range of generative tasks. We extend the discrete modeling paradigm by introducing a particle-level generative model trained on residual-quantized full-event data. We demonstrate the ability of this model family to perform conditional generation from detector-stable particles; we study its scaling behavior across a range of dataset and model sizes, characterize the effects of repeated data exposure and demonstrate that token-level loss systematically predicts downstream physical fidelity. These results provide an empirical framework for scalable collider full-event generation based on residual-quantized representations.

Large language models are characterized by three key properties: capability, alignment, and faithfulness. Prior work studies the tradeoffs between capability and alignment, and between capability and faithfulness, but a third tension remains underexplored: the alignment-faithfulness conflict. We show that aligned models systematically deviate from their inputs on unsafe or sensitive content without disclosing the modification, a failure mode we call alignment-induced unfaithfulness (AIU). Unlike capability-driven unfaithfulness, which comes from errors in knowledge or reasoning, this is induced by post-training mechanisms that override adherence to the input. We introduce FaithConflict, a controlled dataset isolating both conflicts, and two complementary taxonomies: behavioral (B1-B8) and chain-of-thought reasoning (C0-C6). Across models, AIU increases with scale and more sharply than capability-driven unfaithfulness, a reverse scaling law; intermediate checkpoints show it is amplified during post-training, with DPO the stage at which the gap both grows most and becomes least visible. Prompting-based mitigation does not resolve it, revealing a capability-alignment-faithfulness trilemma in the design and evaluation of LLMs.

We study algorithms for detecting induced subgraphs corresponding to fixed pattern graphs in host graphs. We show that at least five of the 21 connected graphs on five vertices can be detected in time roughly the product of the number of vertices and the number of edges, and that at least 65 of the 112 connected graphs on six vertices can be detected in time nearly quadratic in the number of edges. We also give algorithms for detecting induced paths and cycles on seven vertices, running in time roughly the number of vertices times the square of the number of edges. Our main technical tool is a generalized notion of tree decomposition width, called (p, q)-width. It yields algorithms whose running times depend on both the number of vertices and the number of edges, and are never worse than existing bounds. Whenever the host graph has fewer than roughly quadratically many edges in its number of vertices, our bounds are strictly faster. For some patterns, including the seven-vertex cycle, our algorithms are optimal under standard complexity-theoretic assumptions. We further develop this approach using pattern-based polynomials that exploit the structure of tree decompositions, not just their width. This gives algorithms for detecting induced paths and cycles on an even number of vertices in bipartite graphs, running in time roughly the (k-1)-th power of the number of edges for paths on 2k vertices, and that same bound times the number of vertices for cycles on 2k vertices. These are faster than the best known algorithms for general graphs.

Probabilistic record linkage identifies records belonging to the same entity even when identifiers are incomplete or imperfect, and is increasingly used to conduct population-level inferences from administrative data at scale. Linkage produces noisy pairwise links that must then be resolved into distinct individuals. The standard approach, transitive closure, applies a fixed threshold to each link independently and assigns clusters from the resulting connected components. However, the risk of merging distinct individuals grows with population size, motivating approaches that use graph structure to improve linkage accuracy. Clustering algorithms from adjacent fields solve the same graph-clustering task in other domains, yet have never been systematically evaluated for entity resolution. We evaluated eleven clustering algorithms – five established in entity resolution and six from adjacent fields – across three simulated datasets and one real-world dataset, and propose an adaptation tailored to linkage graphs. Cross-field algorithms – Leiden, Louvain, label propagation – were among the most accurate, reducing linkage error by up to 66% relative to transitive closure. The three algorithms also ran up to 40 times faster than Markov clustering (a leading approach in the entity resolution literature) and integrate directly into existing pipelines at scale.

When non-adiabatic coupling mixes electronic states of a molecule, vibronic states of different electronic symmetry become interleaved across the same spectral region. The symmetry of these states cannot be accessed in conventional spectroscopy due to orientation averaging. We show that molecular-frame angular signatures, which are preserved even when states with different symmetry overlap in energy, can be used to resolve them. Using vacuum-ultraviolet transient absorption spectroscopy of aligned carbon dioxide molecules, we separate two electronically distinct classes of vibronic states ($A^{'}$ and $A^{''}$) that arise from Renner-Teller splitting of the $^1\Delta_u$ state. We reconstruct molecular-frame anisotropy continuously across the vibronic manifold showing that the anisotropy crosses over from perpendicular to parallel transition character and maps the interleaved electronic symmetry landscape. By scanning the femtosecond delay between the VUV and a near-infrared pulse through temporal overlap, we measure symmetry-dependent coherence dephasing times of these states and find that the parallel ($A^{'}$) states dephase faster than the perpendicular ($A^{''}$) states. We attribute this to faster departure of the $A^{'}$ wavepacket from the Franck-Condon region due to the distinct topology of the $A^{'}$ excited-state surface. These measurements probe symmetry-resolved ultrafast wavepacket dynamics on the coupled excited-state potential energy surfaces.

Operator learning on probability measures can be accomplished with transformers. For measures with polynomial tails, the exponential weighting in softmax can make the corresponding measure-level attention integrals diverge. This motivates replacing the exponential with slower-growing functions. We construct two benchmarks for operator learning on measures with closed-form targets. We use these benchmarks to study attention kernel growth and data transformation in post-norm transformers. Without data transformation, the softmax models exhibit ensemble collapse on both heavy-tailed benchmarks, while the three slower-growing kernels avoid collapse. Symlog preprocessing allows softmax to avoid collapse on the matrix inverse task but not on the sheared swap task. On the Gaussian control, all four kernels perform similarly. We also examine how sample size affects the sensitivity of empirical energy and Wasserstein distances to tail differences. These results support slower-growing attention kernels as an effective design choice for post-norm transformers learning from heavy-tailed ensembles.

This paper presents a preliminary study of an alternative to the affine transformation underlying conventional neural-network layers. In the proposed Soft Dominance Layer, each output unit compares input coordinates with a learnable reference vector and aggregates smooth inequality responses. A sigmoid relaxation makes the comparisons differentiable, while a sharpness parameter $\alpha$ controls their transition toward hard threshold decisions. The aim is to examine the trainability and direct threshold interpretation of this primitive, not to claim a replacement for affine layers. In single-run MNIST experiments, the highest observed Soft Dominance accuracy is $0.9061$ without annealing and $0.9173$ with annealing, compared with $0.9827$ for the MLP baseline. These descriptive results do not establish reliable configuration rankings or a statistically supported annealing benefit. Learned reference vectors exhibit spatial structure, providing qualitative evidence of structured learning. Repeated-seed experiments and broader datasets are required to assess robustness and practical relevance beyond this proof of concept.

Longitudinal clinical reasoning requires large language models (LLMs) to identify and integrate relevant evidence distributed across extended patient histories. Although long-context models can process increasingly large amounts of information, providing more history does not necessarily make relevant evidence more accessible or improve reasoning. We compare five context strategies (Full, Recent, Episodic, Semantic, and Hybrid) on MedLoCoMo across four open-weight LLMs, examining answer correctness, robustness to query-evidence distance, and abstention on questions with unsupported premises. Episodic and Hybrid generally achieve the strongest overall accuracy, while Recent Context degrades most as supporting evidence becomes more distant; Episodic and Hybrid maintain the highest accuracy at long distances. Analysis of adversarial questions further shows that strong performance on answerable questions does not necessarily translate to successful abstention when the available history does not support the requested conclusion. These findings show that reliable longitudinal reasoning depends not only on how much history an LLM can access, but critically on how relevant evidence is selected and presented for reasoning.

Photon loss is a dominant noise process in bosonic quantum hardware, including superconducting cavities. Fixed-total-photon-number qubit encodings in two bosonic modes retain the loss-detection advantage of dual-rail qubits while supporting photon loss correction. Optimizing entanglement fidelity in this setting for $4\leq n\leq25$ over arbitrary encoders and decoders reveals finite-group structure in every best-found code: 11 correspond to two-dimensional irreducible representations and 11 to reducible ones. Motivated by this emergence, we derive the necessary-and-sufficient Knill-Laflamme conditions for arbitrary finite-group-invariant codes, reducing their construction to equations on representation multiplicity spaces. For every finite subgroup of $SU(2)$ and every tensor rank, we explicitly construct the minimum number of operators required to impose all corresponding loss-correction constraints. These results produce analytic counterparts to most numerical codes and predict constructions missed by the initial search. Exact MacWilliams-Farkas certificates show that our codes achieve the maximum possible loss distance in 21 of the 22 sectors, and we prove that the resulting distance bound is monotone in total photon number. Beyond the scan, we identify a binary-polyhedral sequence with photon number $n_d=\lceil(3d^2+1)/4\rceil$ and construct each corresponding code through distance $d=10$. To our knowledge, the constructed $(n,d)=(28,6),(49,8),(76,10)$ codes give the smallest reported $n$ for their respective distances; all members through $d=9$ attain the fixed-$n$ LP distance bound. Together, these constructions, certificates, and a symmetry-reduced constraint count provide evidence for an infinite code family conjectured to attain every distance at the minimum photon number.

A domain in the Riemann sphere is called a circle domain if every connected component of its boundary is either a round circle or a single point. A circle domain is conformally rigid if every conformal map onto another circle domain is the restriction of a Möbius transformation. Conformal rigidity is closely related to Koebe's conjecture, which asserts that every domain in the Riemann sphere is conformally equivalent to a circle domain. In this article, we survey recent progress on conformal rigidity, with particular emphasis on its relationship with conformal removability.

Vision-Language Models (VLMs) have shown strong multimodal reasoning capabilities, yet whether they truly capture the physical consistency underlying real-world dynamics remains unclear. Existing benchmark paradigms often suffer from fragmented evaluation, focusing on isolated cognitive stages while overlooking the inherent synergy between perception, reasoning, and physical judgment. The lack of a holistic perspective limits the ability to diagnose whether VLMs can reliably serve as evaluators of physical plausibility in generative content. To address these issues, we introduce PhysVista, a benchmark designed to evaluate physical intelligence in VLMs through a closed cognitive loop framework inspired by the human seeing-reasoning-assessment process. PhysVista restores this loop by jointly evaluating physical state perception, physical dynamics reasoning, and physical plausibility assessment. It further distinguishes event-level reasoning and scale-level reasoning to enable fine-grained analysis of physical understanding. In addition, PhysVista incorporates both real-world and AI-generated videos, allowing evaluation across diverse domains and emerging generative scenarios. Extensive experiments across a diverse set of VLMs reveal substantial limitations in physical reasoning and plausibility assessment, highlighting a persistent gap between visual recognition and genuine physical understanding, and pointing toward more principled designs for physically grounded multimodal intelligence.

While Mixture-of-Experts (MoE) models effectively scale model capacity through sparse activation, their deployment is often bottlenecked by prohibitive memory requirements. Extracting a compact subset of experts presents a promising solution. However, existing expert selection heuristics predominantly rely on Top-$k$ ranking, which isolates the evaluation of individual experts and ignores the intricate inter-expert dependencies introduced by the MoE gating network. In this paper, we propose DS-MoE, a theoretically grounded framework that redefines expert selection via difference-of-submodular (DS) optimization. By analyzing the second-order Taylor expansion of the loss degradation, we reveal functional duality within expert combinations: redundancy (where experts encode overlapping representations) and synergy (where experts provide complementary error cancellation). To navigate this duality, we mathematically decouple redundancy reduction from synergy maximization by formulating the selection objective as a DS function. Furthermore, we devise a tailored majorization-minimization (MM) algorithm with provable guarantees to efficiently identify a high-quality expert subset. Extensive experiments demonstrate that DS-MoE effectively preserves indispensable expert combinations, achieving superior performance compared to the state-of-the-art baselines.

Autonomous red team agents increasingly stress-test AI-enabled cyber defenses by planning strategy and executing multistage attacks. Reinforcement learning (RL) and large language models (LLMs) offer complementary mechanisms for the planning and execution such agents require, and prior work has combined them in hybrid hierarchies. Yet a given architecture is typically developed and evaluated within a single environment, leaving open whether an observed advantage reflects a generally stronger decision mechanism or merely alignment with a particular setting. We address this gap with a controlled cross-environment comparison of two homogeneous hierarchical red team architectures: an RL planner with an RL executor (RL+RL) and an LLM planner with an LLM executor (LLM+LLM). We evaluate both against expert autonomous defenders in CybORG CAGE-4 and in Cyberwheel at two network scales, across 18 configurations under one unified disruption metric. We find a pronounced environment-dependent inversion. RL+RL wins the compact, densely rewarded CAGE-4 (78.5% disruption success versus 18.0% for the strongest LLM configuration) and the 100-host Cyberwheel network (81.0% versus 50.5%), while a pretrained cybersecurity LLM agent wins the larger, escalation-gated 1010-host Cyberwheel network (55.0% versus 0.0% for RL). A kill-chain analysis explains the inversion through architecture-specific bottlenecks that aggregate success rates conceal.In the 1010-host Cyberwheel network, RL discovers and compromises hosts but stalls at privilege escalation, whereas in CAGE-4, LLM agents obtain privileged access but rarely convert it into operational impact. These results indicate that conclusions drawn in a single environment may not generalize, and that hybrid planner-executor designs should be motivated by specific failure modes rather than the assumption that one architecture is universally preferable.

In this work we explores the dynamic solutions of a brane system in a five dimensional space-time under the Hamiltonian perspective, using the De Donner Weyl formalism. We obtain the dynamic equations for the scale factors, and obtain the Friedmann-like equations in general way. We present the solutions of the scale factors as a function of the extra coordinate, $y$, we show that the geometry is related to the matter-energy component in the brane. Finally, we develop the ADM formalism in the brane point of view with the foliation of the 3-dimensional space which is embedded in a 5-dimensional manifold.

Software engineering workflows are often not designed to accommodate the needs of developers with Attention Deficit Hyperactivity Disorder (ADHD), despite known challenges related to task initiation, sustained attention, and completion. At the same time, large language models (LLMs) are increasingly integrated into programming tools, but existing systems do not account for neurodiversity or support structured progression across development tasks. In this work, we present Tether 2.0, an LLM based assistant designed to support software engineers with ADHD through workflow oriented interaction across planning, coding, debugging, and review. The tool was named Tether 2.0 because it builds upon the open source code and foundations of the original Tether system. Our approach combines structured interaction modes, activity aware context, and persistent memory to support task progression and continuity. We evaluate the system through expert feedback and hands on use with a software engineer with ADHD. Our results indicate that Tether 2.0 supports requirement clarification, task decomposition, incremental implementation, debugging, and completion, while enabling users to maintain progress and resume work after interruptions. These findings suggest that LLM based assistants can support direction, sustained progress, and learning in software development when designed with workflow structure and context aware interaction.

We evaluate hybrid quantum-classical machine learning for the reduced-order prediction of spatiotemporal brain deformation fields. To mitigate the computational intractability of high-dimensional displacement fields, we employ Proper Orthogonal Decomposition (POD) to project the data into a compact latent space. Within this framework, we formulate two distinct learning objectives: static temporal-to-latent regression and autoregressive latent state forecasting. We systematically benchmark compact classical baselines against both minimal and enhanced hybrid quantum architectures. Our results demonstrate that classical networks provide the strongest baselines in the present setting. For static regression, a classical POD-MLP outperforms all evaluated quantum variants, although an enhanced Variational Quantum Circuit (VQC) substantially improves upon a minimal VQC baseline. For temporal forecasting, a classical POD-LSTM delivers superior predictive accuracy and statistical robustness compared to an enhanced Quantum LSTM (QLSTM) across varying history windows and random initializations. Overall, this study establishes reduced-order physical field learning as a rigorous testbed for near-term QML, highlighting that while hybrid enhancements successfully recover expressivity in weak quantum circuits, classical architectures retain a definitive advantage in both fidelity and stability.

White matter hyperintensities (WMH) and ischaemic stroke lesions (ISL) are visually confounding, co-occurring pathologies that require large, diverse datasets for robust deep learning segmentation. However, assembling such datasets is hindered by cohort samples that lack reference segmentations from both features and complete sets of MRI structural sequences (i.e., "modalities"). To maximise data utility, we investigate hetero-modal learning using a dataset of 206 vascular disease patients across four MRI sequences (T1-weighted, T2-weighted, fluid-attenuated inversion recovery, and diffusion-weighted imaging) with expert annotations of both WMH and ISL. We demonstrate that hetero-modal learning outperforms models trained using a single imaging modality in scenarios with substantial missing data, including a split where only 10% of the training data contains all four modalities while the remainder is uni-modal, and a split relying on a single shared "anchor" modality with zero overlap between the remaining modalities. Furthermore, models trained under this second split successfully perform inference on unseen combinations of modalities. Yet, while standard hetero-modal networks handle missing sequences, clinical deployment introduces the additional challenge of silent data degradation - where modalities are present but severely corrupted. To bridge this gap, we introduce the Multimodal Attention Router (MMAR) block. Our experiments demonstrate that, when trained with a "corruption augmentation" strategy, the MMAR effectively dynamically weights the encoded features of each modality, maintaining strong performance during hetero-modal inference even in the presence of unflagged catastrophically corrupted modalities.

We introduce a conservative two-exponent dispersive family, with $p>1$ the nonlinear-flux exponent and $r>0$ the gradient-dispersion exponent, that separates the mechanisms governing compactification, free-boundary regularity and linearised spectral structure while retaining an explicit travelling-wave theory. The family reduces to the absolute Rosenau-Hyman equation when $r=1$. The deformation leaves the amplitude-speed law unchanged, but makes the width amplitude dependent except at $r=1$. Two independent transition curves emerge: $pr=1$ separates infinite-tail solitary waves from compactons, whereas $r(p-1)=2$ separates compactons with a regular free boundary from edge-cusped compactons with divergent slope. The latter remain distributional solutions, with no singular measures at either the free boundary or the crest. For $r\ge1$, the linearisation factors through a singular Sturm-Liouville operator whose natural Friedrichs realisation is solvable: after a compacton-adapted change of variable, the eigenvalue equation is of Jacobi-hypergeometric type, with two explicit spectral sequences, exactly one negative eigenvalue and a one-dimensional translation kernel. Numerical experiments connect these exact structures to approximation and dynamics. Measured Fourier tails and projection errors agree with the rates predicted by the free-boundary exponent, while padded Fourier-Galerkin time integration propagates the cusped waves coherently with small phase drift. A structure-adapted Jacobi-Galerkin diagnostic of the full generator finds no detectable real part in the tested truncated point spectra. The family therefore provides an explicit testbed in which geometric singularity, weak admissibility, spectral structure and numerical regularity of compactons can be studied within a single degenerate dispersive model.

A longstanding open problem concerning Ramanujan-Sato series is given by determining an expression for the coefficients $c_{11}(n)$ associated with Ramanujan-Sato series of level 11 with a fixed, nested, finite binomial sum for all nonnegative integers $n$, by direct analogy with the known formulas for the coefficients for lower-level Ramanujan-Sato series (and this is clarified and made explicit in our paper). This open problem was given by Cooper et al. in 2015 [J. Math. Anal. Appl.], in Cooper's 2017 textbook on Ramanujan's theta functions, and in Cooper's 2025 contribution to the Contemp. Math. series. This problem seems to have remained open. We succeed in solving this problem, by introducing and proving a double binomial sum formula for $c_{11}(n)$ that is directly analogous to the coefficient formulas for lower-level Ramanujan-Sato series.

Tests for heteroscedasticity in linear regression lose their level or power in three situations: when the data contain outliers, when the variance is not monotone, and when it changes along a variable outside the regressors, such as time. The proposed tests sort the observations by any chosen variable, split them into three equal parts, fit the regression in each part and compare the largest with the smallest error scale. With least squares fits and normal errors, the ratio follows Hartley's maximum F distribution with three groups exactly. With least trimmed squares fits, the ratio resists outliers spread along the ordering. Its square is approximately a maximum F ratio with effective degrees of freedom, whose limit we derive in closed form from the influence function of the trimmed variance. A factorial simulation covered 72 settings of variance shape, ordering variable, number of regressors, contamination and sample size. The robust test had the highest mean size-adjusted power, 67%, against 45% for the next test; when the variance changed along a regressor, which all tests were given, it led White's test after an outlier screen, 67% against 56%. With heavy-tailed t3 errors its lead was similar, 64% against 35%. Unlike the tests built on the regressors, it can follow a variance changing along time, and it identifies the part of the data where the variance changes. Two published robust versions of the Goldfeld-Quandt test, as implemented from their descriptions, did not hold their nominal level. The R package KOTORY implements the methods.

We study negative edge correlation for two well-known models in statistical physics, the arboreal gas and the $q < 1$ random-cluster model. We show that after the edges $e$ and $f$ are removed, their Rayleigh difference is a crossing contribution minus the covariance of two endpoint-connectivity events. For the arboreal gas on an arbitrary finite graph, we analyze the two leading coefficients of this difference. The first is the classical transfer-current square. If it vanishes, the second has an electrical sum-of-squares formula. This proves negative edge correlation at sufficiently large fugacity whenever one of the two leading coefficients of the Rayleigh difference is nonzero, and characterizes simultaneous equality by an equipotential decomposition. For the random-cluster model on the complete graph $K_n$, we prove negative edge correlation throughout $q<1$ when all edge weights are at least $2$. In the uniform case, we prove something stronger:negative correlation holds when all the weights are equal and at least $1$. In particular, distinct edges in a connected spanning subgraph of $K_n$, weighted by a fixed fugacity, are negatively correlated for every $n$. We also explore what seems to be a general phenomenon: positive correlation of connectivity events. For the arboreal gas, we prove positive correlation for connectivity events at high fugacities. On lattices, the conjectured inequality would make the two-point function supermultiplicative and produce a convex inverse correlation length. Exact computations support the graph and matroid conjectures, including the Seymour–Welsh matroid $\mathcal S_8$, where ordinary edge-negative correlation fails. The results in this paper were derived by the authors without the use of Large Language models. The authors did benefit from using GPT-5 Pro for generating code to test out hypotheses as well as for simplifying the arguments.

Developing classical simulation algorithms for noisy quantum circuits is essential to delineating the limits of quantum advantage. Existing classical sampling approaches for general circuits require circuit depths to grow logarithmically with system size, so that noise drives the global output state close to a trivial state. Here we show that local accumulation of noise at a depth independent of system size is already sufficient. Specifically, we prove that any geometrically local circuit composed of unital operations and interspersed with single-qubit depolarizing noise of strength $p$ after $\Theta(p^{-1} \log p^{-1})$ depth can be approximately sampled from a polynomial-time classical computer. This generalizes existing results that impose anticoncentration assumptions or non-universal gate sets in order to obtain a classical sampler at such shallow depths. Our proof maps the output state of the circuit to a polymer model in statistical mechanics, and combines a convergent cluster expansion with hypercontractivity of the depolarizing channel.

We establish functional aggregate queries (FAQs) as a unifying language for exact classical simulation of quantum circuits. A circuit becomes a sum-product query: factors encode gates, internal wire variables are aggregated, and free boundary variables index transition amplitudes. The central insight is that distinct sources of simulation tractability can be exploited within the same InsideOut evaluation scheme. The query specifies what is computed; the evaluation plan, semiring, and representation of intermediate factors determine the cost. This view unifies structural and algebraic simulation guarantees. With explicit factor representations, FAQ evaluation recovers the treewidth bound for tensor-network contraction and yields finer sparsity-sensitive bounds via fractional covers. Over a formal phase semiring, compressed intermediate factors recover rank-width-based simulation for compatible quadratic phase representations. For Clifford circuits, affine-quadratic factors are closed under multiplication and marginalization and remain polynomial in size, yielding polynomial-time exact amplitude computation without any bounded-width assumption. Beyond these recoveries, the framework yields a new tractability criterion: tensor layout symmetry width. This parameter combines local cut-rank with separator symmetry through exact tree-tensor representations. We give a constructive evaluation bound and exhibit a circuit family with bounded tensor layout symmetry width but unbounded phase-graph rank-width and circuit line-graph treewidth. These results establish representation-aware FAQ evaluation as a common algorithmic foundation for classical simulation and a systematic route to new tractable regimes.